Analysis of the Lorenz Equations for Large <i>r</i>

Fowler, A. C.

doi:10.1002/sapm1984703215

Cited by 7 publications

(2 citation statements)

References 16 publications

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“…(1.1) to a simple one-dimensional map which, although only formally valid in a restricted parameter regime, does in fact provide a most useful guide to the dynamics for more general values of p and A. The key idea behind the work of this section is similar to that of section 3 with the matching of solutions obtained for slow and fast phases of the motion-however, whereas the stochastic nature of equations (3.3) forced us to match probability distributions, the deterministic equations (4.1) require the slightly more straightforward task of just matching A , B and C. The techniques are related to those used by Fowler (1984) in his study of the Lorenz equations at large r, and by Chapman et a1 (1990) in their analysis of the Rikitake dynamo equations.…”

Section: Derivation Of the Mapmentioning

confidence: 99%

A low-order model of the shear instability of convection: chaos and the effect of noise

Hughes

Proctor

1990

Nonlinearity

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We consider a third-order system of ordinary differential equations aimed at modelling the shear instability of tall thin cells as found, for example, in thermohaline convection. The nonlinear interactions between the zeroth and first harmonics of the convective cells (A and B) fuel the growth of a horizontal shear (C) which, in turn, becomes of sufficient magnitude to destabilise the convective motions. The most interesting feature of the model is that in certain parameter regimes a small amount of random noise has a tremendous influence on the dynamicein others the noise is unimportant and the system displays many typical features of low-order dynamical systems, such as sequences of period-doubling bifurcations. When the decay rates of B and C are comparable, analytic solutions can be found for both regimes; for the former by solving the Fokker-Planck equation describing the probability distribution of A, B and C, for the latter by reducing the equations to a simple one-dimensional map. These analytic results are a considerable aid in understanding the numerical solutions of the equations for general parameter values.

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Section: Derivation Of the Mapmentioning

confidence: 99%

A low-order model of the shear instability of convection: chaos and the effect of noise

Hughes

Proctor

1990

Nonlinearity

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“…[10,11]. In this limit the Lorenz attractor is degenerate into an 8-figure stable limit cycle ( see also in [14]). In the above mentioned works, the 1 r correction has been studied and simple bifurcations of the limit cycle have been observed.…”

Section: Some Remarks Are In Odermentioning

confidence: 97%

Scaling Properties of the Lorenz System and Dissipative Nambu Mechanics

Axenides

Floratos

2014

Chaos, Information Processing and Paradoxical Games

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In the framework of Nambu Mechanics, we have recently argued that Non-Hamiltonian Chaotic Flows in R 3 , are dissipation induced deformations, of integrable volume preserving flows, specified by pairs of Intersecting Surfaces in R 3 . In the present work we focus our attention to the Lorenz system with a linear dissipative sector in its phase space dynamics.In this case the Intersecting Surfaces are Quadratic. We parametrize its dissipation strength through a continuous control parameter , acting homogeneously over the whole 3-dim. phase space. In the extended -Lorenz system we find a scaling relation between the dissipation strength and Reynolds number parameter r . It results from the scale covariance, we impose on the Lorenz equations under arbitrary rescalings of all its dynamical coordinates.Its integrable limit, ( = 0, fixed r), which is described in terms of intersecting Quadratic Nambu "Hamiltonians" Surfaces, gets mapped on the infinite value limit of the Reynolds number parameter ( r → ∞, = 1). In effect weak dissipation, through small values, generates and controls the well explored Route to Chaos in the large r-value regime. The non-dissipative = 0 integrable limit is therefore the gateway to Chaos for the Lorenz system. † The present work is dedicated to the memory of Prof. J.S.Nicolis

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